Confused about perturbation theory with path integrals

[Hazz]

Homework Statement

Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you can find the shift for any other energy level. I am aware you don't need to use a path integral for this but I am hoping to get a better understanding of them by doing this.

Homework Equations

I know that you can calculate the energy shift in the ground state using the partition function by using

E_0=-lim_{\beta\rightarrow\infty}\frac{1}{\beta} log K_E[J] |_{J=0}

I know if I am using the Hamiltonian
\mathcal{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{q}^2+\gamma\hat{q}^4

then to first order you can just write $K_E[J]$ as

K_E[J]=K_E^0[J]-\gamma \int d\tau (\frac{\delta}{\delta J(\tau)})^4K^0_E[J]|_{J=0}

where

K_E^0[J]=\frac{C}{2}\int d\tau d\tau'J(\tau)G(\tau,\tau')J(\tau')

and G is the green's function.

The Attempt at a Solution

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So you can evaluate this all and work it out for the ground state which I don't have a problem with, but what confuses me is how this works for anything that isn't the ground state. Clearly the $\beta\rightarrow\infty$ limit is always going to put you there.

I understand that you probably need to use

\langle 0 |\hat{a}e^{-\beta H} \hat{a}^\dagger |0\rangle

In some capacity but I don't really see where to proceed from there. Do you have to worry about commuting $\hat{a}^\dagger$ through the exponential?

I don't really know where to look for guidance with this sort of thing so any help would be appreciate. Thank you very much!

 

[Hazz]

I found the answer I was looking for section 3.18 of these notes by Hagen Kleinert: http://users.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic03.pdf